Mathwords logoMathwords

Amplitude — Definition, Formula & Examples

Amplitude

Half the difference between the minimum and maximum values of the range. Only periodic functions with a bounded range have an amplitude. Essentially, amplitude is the radius of the range.

 

Sine wave on x-y axes with arrows showing amplitude (half the distance from min to max) labeled on the right.

Key Formula

A=max−min2A = \frac{\text{max} - \text{min}}{2}
Where:
  • AA = Amplitude of the function
  • max\text{max} = Maximum value the function reaches
  • min\text{min} = Minimum value the function reaches

Worked Example

Problem: Find the amplitude of the function y = 3 sin(2x) + 1.
Step 1: Recall that for y = A sin(Bx) + D, the maximum value is D + |A| and the minimum value is D − |A|.
max=1+3=4,min=1−3=−2\text{max} = 1 + 3 = 4, \quad \text{min} = 1 - 3 = -2
Step 2: Apply the amplitude formula: half the difference between maximum and minimum.
A=4−(−2)2=62=3A = \frac{4 - (-2)}{2} = \frac{6}{2} = 3
Step 3: You can also read the amplitude directly from the equation. The coefficient in front of sin is 3, and since amplitude is always positive, the amplitude is |3| = 3.
A=∣3∣=3A = |3| = 3
Answer: The amplitude is 3. The function oscillates 3 units above and 3 units below its midline y = 1.

Another Example

Problem: A Ferris wheel has its highest point at 40 meters and its lowest point at 4 meters above the ground. What is the amplitude of a rider's height as a function of time?
Step 1: Identify the maximum and minimum heights.
max=40 m,min=4 m\text{max} = 40 \text{ m}, \quad \text{min} = 4 \text{ m}
Step 2: Compute the amplitude using the formula.
A=40−42=362=18 mA = \frac{40 - 4}{2} = \frac{36}{2} = 18 \text{ m}
Answer: The amplitude is 18 meters. The rider moves 18 m above and 18 m below the midline height of 22 m.

Frequently Asked Questions

Can amplitude be negative?
No. Amplitude is always a non-negative value because it represents a distance. If you see y = −5 sin(x), the coefficient is −5, but the amplitude is |−5| = 5. The negative sign flips the graph, but the height of the wave is still 5.
How do you find the amplitude from a graph?
First, identify the highest point (maximum) and lowest point (minimum) on the graph. Subtract the minimum from the maximum and divide by 2. Alternatively, find the midline (the horizontal line halfway between max and min) and measure the vertical distance from the midline to the peak — that distance is the amplitude.

Amplitude vs. Period

Amplitude and period describe different aspects of a wave. Amplitude measures the vertical height — how far the function rises and falls from its center. Period measures the horizontal length — how far along the x-axis you travel before the pattern repeats. For y = A sin(Bx), the amplitude is |A| (vertical stretch) and the period is 2π/|B| (horizontal stretch). Changing one does not affect the other.

Why It Matters

Amplitude shows up throughout science and engineering. In physics, the amplitude of a sound wave determines its loudness, and the amplitude of a light wave relates to its brightness. Whenever you model repeating behavior — tides, heartbeats, alternating current, seasonal temperatures — amplitude tells you the size of the oscillation.

Common Mistakes

Mistake: Using the full distance from max to min as the amplitude instead of half that distance.
Correction: Amplitude is half the total vertical span. If a wave goes from −3 to 3, the amplitude is 3, not 6. The full span (max − min) is called the peak-to-peak value.
Mistake: Reporting a negative amplitude because the coefficient in front of sin or cos is negative.
Correction: Amplitude is always non-negative. Take the absolute value of the coefficient: for y = −4 cos(x), the amplitude is |−4| = 4.

Related Terms